I have a Kenmore power miser 12 hot water heater that has a cold water inlet tube based self-cleaning feature. Apparently, flowing cold water pushes the tube around the bottom of the tank, cleaning sediment off the bottom of the tank. Sounds good right?

My faucets are clogged with white mineral debris, and I must clean the faucet screens out almost once a week.

I have a water softener upstream of the hot water heater and the water company has tested the mineral content of the incoming water.

I suspect the hot water heater self cleaning feature is scraping tank sediment into the hot water stream and clogging the faucets. Does this sound plausible?
Is there a cleanable filter I can put inline down stream of the hot water heater to catch the debris?

FYI, the tank drain is plumbed in to cycle hot water for faster heat response

I have a similar setup at my house. The return line is plumbed into the tank drain valve, which provides recirculation and instant hot water at the far end of the house. I modified the attachment of this line to the bottom of the heater so I can attach a hose at the bottom of the tank and flush out the sediment. This works pretty well.

You could plumb one of these in, see hot water filter on bottom of page.